If the area & circumference had the same number value what is the diameter of the circle?
A fraction i equivalent to 2/3. if 5 is added to the numerator than the fraction is 5/6. What was the original fraction?
If y is four less than the square root of x, find x.
The ratio of men to women is 2:3. If there are 365 people, the number of women is
πr^2 = 2πr
r^2 - 2r = 0
r(r-2) = 0
r = 0 , the trivial answer
or
r = 2
So the diameter is 4
a/b = 2/3 or 2b = 3a, or b = 3a/2
(a+5)/b = 5/6
5b = 6a + 30
5(3a/2) = 6a + 30
times 2
15a = 12a + 60
a = 20 and b = 3(20)/2 = 30
the fraction was 20/30
y < √x by 4
y = √x + 4
y-4 = √x
square both sides
y^2 - 8y + 16 = x
men: women = m : w = 2 : 3 = 2x : 3x
5x = 365
x =73
men = 2x = 146
women = 3x = 219
To find the diameter of a circle when the area and circumference have the same value, we need to use the formulas for the area and circumference.
1. Let's start by considering the formula for the area of a circle: A = πr^2, where A is the area and r is the radius.
2. Next, let's consider the formula for the circumference of a circle: C = 2πr, where C is the circumference.
3. Since we know that the area and circumference have the same value, we can set up an equation: A = C.
4. Substitute the formulas for area and circumference into the equation: πr^2 = 2πr.
5. Divide both sides of the equation by π to simplify: r^2 = 2r.
6. Subtract "2r" from both sides of the equation and rearrange the terms: r^2 - 2r = 0.
7. Factor out an "r" from the left side of the equation: r(r - 2) = 0.
8. Set each factor equal to zero and solve for r: r = 0 or r - 2 = 0.
9. Since the radius cannot be zero (as it wouldn't create a circle), we can ignore the solution r = 0.
10. Solve for r - 2 = 0: r = 2.
11. Finally, to find the diameter, we multiply the radius by 2: diameter = 2 * r = 2 * 2 = 4.
Therefore, the diameter of the circle when the area and circumference have the same value is 4.
For the second question,
1. Let's assume the original fraction is "x/y."
2. According to the given information, the equivalent fraction with 5 added to the numerator is "x + 5/y."
3. We are told that this equivalent fraction is 5/6.
4. Set up an equation using the given information: (x + 5)/y = 5/6.
5. Cross-multiply to get rid of the fractions: 6(x + 5) = 5y.
6. Distribute the 6 on the left side: 6x + 30 = 5y.
7. Rearrange the equation to isolate "x": 6x = 5y - 30.
8. Divide both sides of the equation by 6: x = (5y - 30)/6.
Now, to find the original fraction, we need to substitute the value of "x + 5" back into the equation:
9. Replace "x + 5" with the equivalent fraction: (5y - 30)/6 + 5 = 5/6.
10. Multiply through by 6 to eliminate the fractions: 5y - 30 + 30 = 5.
11. Simplify the equation: 5y = 5.
12. Divide both sides by 5: y = 1.
13. Substitute the value of y back into the equation for "x": x = (5(1) - 30)/6.
14. Calculate: x = (-25)/6.
Therefore, the original fraction is (-25)/6.
For the third question,
1. Let's represent the given information using variables: y = √x - 4.
2. We want to find the value of x.
3. Square both sides of the equation to eliminate the square root: (y)^2 = (√x - 4)^2.
4. Simplify the equation: y^2 = (x - 8√x + 16).
5. Expand the equation: y^2 = x - 8√x + 16.
6. Rearrange the terms to isolate the square root: 8√x = x - y^2 + 16.
7. Square both sides again to eliminate the square root: (8√x)^2 = (x - y^2 + 16)^2.
8. Simplify: 64x = x^2 - 2xy^2 + y^4 - 32x + 256.
9. Rearrange the equation to isolate x: 0 = x^2 - 64x - 2xy^2 + y^4 - 32x + 256.
10. Combine like terms: 0 = x^2 - 96x - 2xy^2 + y^4 + 256.
11. Factor the equation if possible. If not, use the quadratic formula to find the values of x.
12. Solve for x using either the factored equation or the quadratic formula.
Unfortunately, the equation cannot be easily factored, so we need to use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a).
In this case, a = 1, b = -96, and c = (-2y^2 + y^4 + 256).
Plug in these values and solve for x.
Therefore, the value of x depends on the value of y, and without knowing the numerical value of y, we cannot find the specific value of x.
For the fourth question,
1. Let's assume the ratio of men to women is 2:3.
2. The total number of people in the ratio is 2 + 3 = 5.
3. The fraction that represents the number of women would be 3/5, as there are 3 parts out of 5.
4. We are told that there are 365 people in total.
5. Multiply the fraction by the total number of people to find the number of women: (3/5) * 365 = 219.
Therefore, the number of women is 219.